universality of tensor products

DEFINITION 09.2   Let $A$ be a (not necessarily commutative) ring. Let $M$ be a right $A$-module. Let $N$ be a left $A$-module. Then for any module $X$, a map $f:M\times N \to X$ is said to be an $A$-balanced biadditive map if it satisfies the following conditions.

1. $f(m_1+m_2,n)=f(m_1,n)+f(m_2,n) \quad (\forall m_1,m_2 \in M, \forall n \in N)$
2. $f(m,n_1+n_2)=f(m,n_1)+f(m,n_2) \quad (\forall m \in M, \forall n_1,n_2 \in N)$
3. $f(m a ,n)=f(m, a n) \quad (\forall m \in M, \forall n \in N, \forall a \in A)$

LEMMA 09.3   Let $A$ be a (not necessarily commutative) ring. Let $M$ be a right $A$-module. Let $N$ be a left $A$-module. Then for any module $X$, there is a bijective additive correspondence between the following two objects.

1. An $A$-balanced bilinear map $M\times N \to X$
2. An additive map $M\otimes_A N \to X$

Universality argmuments are deeply related to the uniqueness of initial objects. Consult Lang “Algebra”.